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# The symbol ∆ denotes one of the four arithmetic operations:addition,su

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22 Jun 2017, 15:18
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The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
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Updated on: 29 Jul 2019, 09:32
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AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations: addition,subtraction,multiplication,or division. If 6∆3≤ 3, which of the following must be true ?

I.2∆2=0
II.2∆2=1
III.4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

If 6 ∆ 3 ≤ 3, then the ∆ operator is subtraction AND division.

6 - 3 = 3, which satisfies the equation

$$\frac{6}{3}$$ = 2, which also satisfies the equation (2 is less than 3).

The operator cannot be addition or multiplication. 6 + 3 = 9, which is > 3. And 6 * 3 = 18, also > than 3.

Check I, II, and III.

I. 2 ∆ 2 = 0. If ∆ were only subtraction, this would be fine. 2 - 2 = 0.

But $$\frac{2}{2}$$ = 1. This equation is not true when ∆ is division. Incorrect.

II. 2 ∆ 2 = 1. If ∆ were only division, this would be fine.$$\frac{2}{2}$$ = 1. But when ∆ is subtraction, then 2 - 2 = 0. This equation does not hold true when ∆ is subtraction. Incorrect.

III. 4 ∆ 2 = 2. This must be true for both subtraction and division.

It holds for subtraction. 4 - 4 = 2
And it holds for division: $$\frac{4}{2}$$= 2. Correct.

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Originally posted by generis on 22 Jun 2017, 16:43.
Last edited by generis on 29 Jul 2019, 09:32, edited 3 times in total.
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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22 Jun 2017, 16:49
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1
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I.2∆2=0
I I .2∆2=1
I I I .4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

$$6∆3≤ 3$$

$$\frac{6}{3}= 3$$

$$6-3 = 3$$

Therefore symbol ∆ should be division and subtraction both.

Checking the options, we get;

I.$$2∆2=0$$

$$2 - 2 = 0$$

$$\frac{2}{2}= 1$$------------ Must Not be true.

II .$$2∆2=1$$

$$\frac{2}{2}= 1$$

$$2 - 2 = 0$$ --------------- Must Not be true.

III .$$4∆2=2$$

$$\frac{4}{2} = 2$$ --------- Must be true.

$$4 - 2 = 2$$ --------- Must be true.

Hence Only III must be true. Answer (C)...
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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23 Jun 2017, 00:59
1
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I.2∆2=0
I I .2∆2=1
I I I .4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

∆ could be either Subtraction or Division in order to fulfill condition 6∆3≤ 3
I - 2∆2=0 - In this condition, ∆ denotes only subtraction and not Division. Hence Wrong
II - 2∆2=1 - In this condition, ∆ denotes only Division and not subtraction. Hence Wrong
III - 4∆2=1 - In this condition, ∆ denotes either Division or subtraction. Hence Correct.

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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23 Jun 2017, 01:24
From the context, $$∆= subtraction and division$$
∆= subtraction$$6-3<=3$$
∆= division$$\frac{6}{3}=2<=3$$

From the options
Only iii holds true as 4∆2=2( both division and subtraction)

Option c
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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26 Jun 2017, 17:00
2
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I.2∆2=0
I I .2∆2=1
I I I .4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Let’s analyze the given inequality:

6 + 3 ≤ 3

9 ≤ 3

Since 9 is not less than 3, ∆ cannot be addition.

If ∆ = subtraction:

6 - 3 ≤ 3

3 ≤ 3

Since 3 is less than or equal to 3, ∆ can be subtraction.

If ∆ = divison:

6/3 ≤ 3

2 ≤ 3

Since 2 is less than or equal to 3, ∆ can be division.

If ∆ = multiplication:

6 x 3 ≤ 3

18 ≤ 3

Since 18 is not less than or equal to 3, ∆ cannot be multiplication.

We see then that ∆ is either subtraction or division. Now let’s analyze each Roman numeral:

I. 2∆2=0

Since 2/2 does not equal 0, division doesn’t work and I is not true.

II. 2∆2=1

Since 2 - 2 does not equal 1, subtraction doesn’t work and II is not true.

III. 4∆2=2

Since 4 - 2 = 2 and 4/2 = 2, III is true.

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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27 Jun 2017, 14:49
1
Top Contributor
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2=0
II. 2∆2=1
III. 4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

The above solutions are perfect.
I just want to point out that, when answering this type of question type (where the answer choices are I only, II and III only, etc.), be sure to ELIMINATE answer choices after examining each statement.

So, for example, once we determine that statement I need not be true, we can ELIMINATE answer choices A, D and E
Then, once we determine that statement II need not be true, we can ELIMINATE answer choice B

At this point, there's no need to check statement III, since we already know that the correct answer must be C

Cheers,
Brent
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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10 Jul 2017, 11:43
1
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I.2∆2=0
I I .2∆2=1
I I I .4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

III (.4)∆2=2

and got it all wrong!!

That dot, right after the Roman Numerals, is so confusing...!
The dot is closer to the digit, rather, it should be closer to the Roman numeral, to avoid any confusion! So I suggest, this question must be edited!

Thanks,
ashygoyal
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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10 Jul 2017, 11:48
ashygoyal wrote:
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I.2∆2=0
I I .2∆2=1
I I I .4∆2=2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

III (.4)∆2=2

and got it all wrong!!

That dot, right after the Roman Numerals, is so confusing...!
The dot is closer to the digit, rather, it should be closer to the Roman numeral, to avoid any confusion! So I suggest, this question must be edited!

Thanks,
ashygoyal

Edited as suggested. Thank you.
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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10 Jul 2017, 22:21
1
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

6∆3≤ 3

So, 6 - 3 = 3
6/3 = 2

I. 2∆2
2-2 = 0
2/2 = 1
So it may be true.

II. 2∆2
2-2 = 0
2/2 = 1
So it is may be true.

III. 4∆2
4-2 = 2
4/2 = 2
So, 4∆2 = 2 must be true.

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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11 Jul 2017, 00:06
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

6∆3≤ 3
∆ can denote subtraction or division .
6+3=9 and 9> 3
6*3 = 18 and 18 >3

I. 2∆2 = 0 , If ∆ is division , then the expression is invalid

II. 2∆2 = 1 , If ∆ is subtraction , then the expression is invalid
III. 4∆2 = 2 - Correct

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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21 Aug 2017, 21:40
1
Thinking that even one criteria satisfies the condition, I have interpreted that all three options are valid

Since the question says 6∆3=<3
6/3 = 2
6-3 = 3
I applied the conditions individually for all three and opted for E.
But didn't strike that one of the option should satisfy both conditions.
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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09 Jan 2018, 17:39
Hi All,

We're told that the symbol # denotes one of the four arithmetic operations: addition, subtraction, multiplication, or division and that 6#3 <= 3. We're asked which of the three Roman Numerals must be true.

To start, let's do a bit of analysis on the inequality that we're given (and determine which math operation COULD fit this):
6#3 <= 3

6 + 3 = 9 which is NOT <= 3
6 - 3 = 3 which IS <= 3
(6)(3) = 18 which is NOT <= 3
6/3 = 2 which IS <=3

Thus, the symbol # must be either subtraction or division (although we don't know which one for sure).

I. 2#2 = 0

We've proven that the symbol is either subtraction or division, so let's see if either of those operations 'fits' this information...
2 - 2 = 0
2/2 = 1
Thus, if the symbol is subtraction, then Roman Numeral 1 IS true. However, if the symbol is division, then Roman Numeral 1 is NOT true. There's no way to know which symbol is involved though, so Roman Numeral 1 is not necessarily true.
Eliminate Answers A, D and E.

II. 2#2 = 1

With the work that we've done in Roman Numeral 1 (above), we already know that the outcome of 2#2 could be 0 or 1. This inconsistency also provides that Roman Numeral 2 is not necessarily true.

III. 4#2 = 2

You can still prove that Roman Numeral 3 is always true...
4 - 2 = 2
4/2 = 2
Both outcomes 'fit' the information in Fact 3, so regardless of what operation is represented, the statement IS true.

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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14 Jan 2019, 09:59
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Given 6∆3≤ 3, this means ∆, can either be a "-" or a "/"
I. 2∆2 = 0, 2 - 2 = 0 But 2 / 2 != 0
II. 2∆2 = 1, 2 - 2 != 1 But 2 / 2 = 1
III. 4∆2 = 2, 4 / 2 = 2 & 4 - 2 = 2

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14 Jan 2019, 10:03
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

If ∆ represents + then 6∆3≤ 3 = 6 + 3 ie 9 > 3 rejected

If ∆ represents - then 6∆3≤ 3 = 6 - 3 ie 3 = 3 Accepted

If ∆ represents ÷ then 6∆3≤ 3 = 6 ÷ 3 ie 2 < 3 Accepted

Now, check the given options -

(I) 2∆2 = 0 , if ∆ represents ÷ then 2∆2 = 1 (rejected)

(II) 2∆2 = 1 , if ∆ represents - then 2∆2 = 0 (rejected)

(III) 4∆2 = 1 , if ∆ represents - then 4∆2 = 2 (Accepted) ; if ∆ represents ÷ then 4∆2 = 2 (Accepted)

Thus, Answer must be (C) Only option (III)
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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14 Jun 2019, 00:09
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Why does the question say "denotes one of the four arithmetic operations", wherein while solving, ∆ actually denotes two operations (both subtraction and division).

Coming from the makers of the GMAT itself, isn't the language a bit unclear?
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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14 Jun 2019, 13:41
rishabhjain13 wrote:
AbdurRakib wrote:
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Why does the question say "denotes one of the four arithmetic operations", wherein while solving, ∆ actually denotes two operations (both subtraction and division).

Coming from the makers of the GMAT itself, isn't the language a bit unclear?

Hi rishabhjain13,

Most GMAT questions (in both the Quant and Verbal sections) require 3-5 'steps' to solve - and in many cases, at least one of the steps requires that you properly 'link' one piece of information to another and make a deduction. As a simple example, you might see a question that states A and B are integers and B > 0. The deduction here is that A can be any type of integer (negative, 0 or positive), while B can only be a positive integer. You might be thinking "why not just say that B is a positive integer?".... but that removes one of the essential elements of the question (and of the overall Exam): measuring how well you can link ideas and make deductions.

The same concept applies to this question. We start off with a broader piece of information (re: there are 4 possible math functions, but which one is it?) and then we're given a second piece of information that narrows down the possibilities (re: it's either subtraction or division, but we still don't know exact which one until we move on to the next part of the question).

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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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15 Jul 2019, 00:11
The symbol ∆ denotes one of the four arithmetic operations:addition,subtraction,multiplication,or division,If 6∆3≤ 3 ,which of the following must be true ?

I. 2∆2 = 0
II. 2∆2 = 1
III. 4∆2 = 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

6/3 <=3 , 6-3 <=3 same in opt 3 4/2 <=2 , 4-2<=2
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Re: The symbol ∆ denotes one of the four arithmetic operations:addition,su  [#permalink]

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16 Jul 2019, 23:30
Given:6∆3≤3

Checking for all signs

6-3=3
6/3=2
6*3=18
6+3=9
So, ∆ can be division and subtraction.

Checking the options, we get;

I.2∆2=0
2−2=0 - Correct
2/2=1 - Incorrect

II .2∆2=1

2/2=1 - Correct
2−2=0 - Incorrect

III .4∆2=24∆2=2

42=2 - Correct
4−2=2 - Correct

Hence C
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29 Jul 2019, 17:14

As a test taker -- how are we supposed to know that ∆ should represent BOTH division AND subtraction

I assumed, the conditions applied individually (subtraction OR division) and thus opted for E

It's clear one or the other works

Since the question says 6∆3=<3
6/3 = 2
OR
6-3 = 3

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The symbol ∆ denotes one of the four arithmetic operations:addition,su   [#permalink] 29 Jul 2019, 17:14

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